2302:
4061:; over an algebraically closed field every semisimple Lie algebra is splittable. Any two splitting Cartan algebras are conjugate, and they fulfill a similar function to Cartan algebras in semisimple Lie algebras over algebraically closed fields, so split semisimple Lie algebras (indeed, split reductive Lie algebras) share many properties with semisimple Lie algebras over algebraically closed fields.
35:
1750:
4083:
is a special type of subgroup. Specifically, its Lie algebra (which captures the groupâs algebraic structure) is itself a Cartan subalgebra. When we consider the identity component of a subgroup, it shares the same Lie algebra. However, there isnât a universally agreed-upon definition for which
1591:
2775:
1586:
3064:
950:
if it consists of semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over an algebraically closed field of characteristic zero, a Cartan subalgebra can also be defined as a maximal toral subalgebra.
1885:
4106:. This version is sometimes called the âlarge Cartan subgroup.â Additionally, there exists a âsmall Cartan subgroup,â defined as the centralizer of a maximal torus. Itâs important to note that these Cartan subgroups may not always be abelian in genera
2871:
3411:
2657:
2483:
3531:
1453:
933:
3182:
3848:
2281:
matrix. One can directly see this abelian subalgebra is not a Cartan subalgebra, since it is contained in the nilpotent algebra of strictly upper triangular matrices (or, since it is normalized by diagonal
3300:
2219:
2668:
1745:{\displaystyle d(a_{1},\ldots ,a_{n})={\begin{pmatrix}a_{1}&0&\cdots &0\\0&\ddots &&0\\\vdots &&\ddots &\vdots \\0&\cdots &\cdots &a_{n}\end{pmatrix}}}
1800:
2562:
2519:
2936:
2044:
2992:
1933:
1795:
1448:
4055:
3448:
3305:
3124:
1082:
For a finite-dimensional complex semisimple Lie algebra, the existence of a Cartan subalgebra is much simpler to establish, assuming the existence of a compact real form. In that case,
3570:
3882:
2784:
1345:
791:
761:
3782:
731:
3808:
3751:
3093:
1225:
is semisimple and the field has characteristic zero, then a maximal toral subalgebra is self-normalizing, and so is equal to the associated Cartan subalgebra. If in addition
4011:
3968:
3921:
3702:
3678:
3618:
3222:
2984:
2960:
2586:
2420:
2393:
2365:
1299:
1275:
1247:
1223:
1199:
1167:
1135:
1104:
1062:
1030:
1002:
822:
685:
658:
2594:
872:
1998:
The dimension of a Cartan subalgebra is not in general the maximal dimension of an abelian subalgebra, even for complex simple Lie algebras. For example, the Lie algebra
1403:
1372:
3590:
3471:
798:
452:
4098:
Now, when we explore disconnected compact Lie groups, things get interesting. There are multiple definitions for a Cartan subgroup. One common approach, proposed by
3943:
2891:
2166:
2139:
500:
2090:
2067:
3723:
3639:
3476:
3243:
2279:
2259:
2239:
2110:
1993:
1973:
1953:
842:
2432:
505:
495:
490:
4194:
4080:
310:
3253:
2525:(As noted earlier, a Cartan subalgebra can in fact be characterized as a subalgebra that is maximal among those having the above two properties.)
574:
457:
886:
4356:
4317:
4283:
4232:
4155:
1250:
605:
1067:
In a finite-dimensional Lie algebra over an algebraically closed field of characteristic zero, all Cartan subalgebras are conjugate under
961:
also have subalgebras that play the same role as the Cartan subalgebras of semisimple Lie algebras (over a field of characteristic zero).
3132:
4258:
3813:
958:
2171:
2770:{\displaystyle {\mathfrak {g}}_{\lambda }=\{x\in {\mathfrak {g}}:{\text{ad}}(h)x=\lambda (h)x,{\text{ for }}h\in {\mathfrak {h}}\}}
1581:{\displaystyle {\mathfrak {h}}=\left\{d(a_{1},\ldots ,a_{n})\mid a_{i}\in \mathbb {C} {\text{ and }}\sum _{i=1}^{n}a_{i}=0\right\}}
467:
974:
3059:{\displaystyle {\mathfrak {g}}={\mathfrak {h}}\oplus \left(\bigoplus _{\lambda \in \Phi }{\mathfrak {g}}_{\lambda }\right)}
4340:
4220:
3651:
2531:
2488:
462:
442:
4199:
2900:
2001:
1893:
1755:
1408:
407:
315:
4020:
4335:
4084:
subgroup with this property should be called the âCartan subgroup,â especially when dealing with disconnected groups.
3986:
3980:
3416:
1064:
is then the same thing as a maximal toral subalgebra and the existence of a maximal toral subalgebra is easy to see.
3248:
2368:
2317:
447:
3098:
1880:{\displaystyle {\mathfrak {h}}=\left\{{\begin{pmatrix}a&0\\0&-a\end{pmatrix}}:a\in \mathbb {C} \right\}}
3302:
there is a decomposition related to the decomposition of the Lie algebra from its Cartan subalgebra. If we set
598:
82:
2866:{\displaystyle \Phi =\{\lambda \in {\mathfrak {h}}^{*}\setminus \{0\}|{\mathfrak {g}}_{\lambda }\neq \{0\}\}}
4102:, defines it as the group of elements that normalize a fixed maximal torus while preserving the fundamental
3536:
3656:
But, it turns out these weights can be used to classify the irreducible representations of the Lie algebra
3854:
3189:
2341:
2292:
1316:
848:
766:
736:
402:
365:
333:
320:
3756:
1004:
over an algebraically closed field of characteristic zero, there is a simpler approach: by definition, a
954:
694:
4375:
1076:
1037:
940:
631:
434:
102:
4330:
3985:
Over non-algebraically closed fields, not all Cartan subalgebras are conjugate. An important class are
3787:
3730:
3072:
3406:{\displaystyle V_{\lambda }=\{v\in V:(\sigma (h))(v)=\lambda (h)v{\text{ for }}h\in {\mathfrak {h}}\}}
2652:{\displaystyle {\mathfrak {g}}=\bigoplus _{\lambda \in {\mathfrak {h}}^{*}}{\mathfrak {g}}_{\lambda }}
3992:
3949:
3890:
3683:
3659:
3599:
3203:
2965:
2941:
2567:
2401:
2374:
2346:
1280:
1256:
1228:
1204:
1180:
1148:
1116:
1085:
1043:
1033:
1011:
983:
881:
803:
666:
639:
62:
52:
2423:
970:
936:
591:
579:
420:
250:
855:
4275:
1138:
351:
341:
4150:. Takeuchi, Kiyoshi, 1967-, Tanisaki, Toshiyuki, 1955- (English ed.). Boston: BirkhÀuser.
4064:
Over a non-algebraically closed field not every semisimple Lie algebra is splittable, however.
1382:
1351:
4352:
4326:
4313:
4279:
4254:
4228:
4161:
4151:
4058:
3575:
3456:
1174:
530:
415:
268:
943:). Sometimes this characterization is simply taken as the definition of a Cartan subalgebra.
876:
a Cartan subalgebra is the same thing as a maximal abelian subalgebra consisting of elements
1005:
947:
688:
550:
230:
222:
214:
206:
198:
177:
167:
157:
147:
131:
112:
72:
4293:
4242:
3928:
2876:
2144:
1141:(a Lie subalgebra of the Lie algebra of endomorphisms of a finite-dimensional vector space
4309:
4289:
4267:
4238:
4224:
4073:
2115:
535:
288:
273:
44:
2478:{\displaystyle \operatorname {ad} :{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})}
2072:
2049:
4087:
For compact connected Lie groups, a Cartan subgroup is essentially a maximal connected
3708:
3624:
3228:
2264:
2244:
2224:
2095:
1978:
1958:
1938:
827:
794:
555:
540:
373:
278:
2301:
4369:
4302:
4092:
4088:
1348:
1107:
263:
92:
4212:
4103:
1068:
560:
545:
346:
328:
258:
2564:
are simultaneously diagonalizable and that there is a direct sum decomposition of
3473:, there is a decomposition of the representation in terms of these weight spaces
4099:
2894:
1170:
1072:
661:
619:
386:
302:
26:
969:
Cartan subalgebras exist for finite-dimensional Lie algebras whenever the base
4095:.â The Lie algebra associated with this subgroup is also a Cartan subalgebra.
2313:
634:
525:
391:
283:
4165:
22:
4145:
3526:{\displaystyle V=\bigoplus _{\lambda \in {\mathfrak {h}}^{*}}V_{\lambda }}
4253:, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
4251:
Lie Groups, Lie
Algebras, and Representations: An Elementary Introduction
928:{\displaystyle \operatorname {ad} (x):{\mathfrak {g}}\to {\mathfrak {g}}}
973:
is infinite. One way to construct a Cartan subalgebra is by means of a
482:
1032:
that consists of semisimple elements (an element is semisimple if the
1145:) over an algebraically closed field, then any Cartan subalgebra of
977:. Over a finite field, the question of the existence is still open.
34:
3177:{\displaystyle \dim {\mathfrak {g}}=\dim {\mathfrak {h}}+\#\Phi }
2521:
consists of semisimple operators (i.e., diagonalizable matrices).
1075:. The common dimension of a Cartan subalgebra is then called the
851:
over an algebraically closed field of characteristic zero (e.g.,
3843:{\displaystyle \langle \alpha ,\lambda \rangle \in \mathbb {N} }
3295:{\displaystyle \sigma :{\mathfrak {g}}\to {\mathfrak {gl}}(V)}
2296:
2214:{\displaystyle {\begin{pmatrix}0&A\\0&0\end{pmatrix}}}
1106:
may be taken as the complexification of the Lie algebra of a
3945:
contains all information about the representation theory of
3646:
Classification of irreducible representations using weights
1301:
is Cartan if and only if it is a maximal toral subalgebra.
3989:: if a Lie algebra admits a splitting Cartan subalgebra
4072:
For a Cartan subgroup of a linear algebraic group, see
3196:
Decomposing representations with dual Cartan subalgebra
2321:
1310:
Any nilpotent Lie algebra is its own Cartan subalgebra.
4304:
Introduction to Lie
Algebras and Representation Theory
4147:
D-modules, perverse sheaves, and representation theory
2180:
1824:
1641:
1376:
over a field, is the algebra of all diagonal matrices.
4023:
3995:
3952:
3931:
3893:
3857:
3816:
3790:
3759:
3733:
3711:
3686:
3662:
3627:
3602:
3578:
3539:
3479:
3459:
3419:
3308:
3256:
3231:
3206:
3135:
3101:
3075:
2995:
2968:
2944:
2903:
2879:
2787:
2671:
2597:
2570:
2534:
2491:
2435:
2404:
2377:
2349:
2267:
2247:
2227:
2174:
2147:
2118:
2098:
2075:
2052:
2004:
1981:
1961:
1941:
1896:
1803:
1758:
1594:
1456:
1411:
1385:
1354:
1319:
1283:
1259:
1231:
1207:
1183:
1151:
1119:
1088:
1046:
1014:
986:
889:
858:
830:
806:
769:
739:
697:
669:
642:
2557:{\displaystyle \operatorname {ad} ({\mathfrak {h}})}
2514:{\displaystyle \operatorname {ad} ({\mathfrak {h}})}
1797:
the Cartan subalgebra is the subalgebra of matrices
4349:
Cohomological
Induction and Unitary Representations
2931:{\displaystyle {\mathfrak {g}}_{0}={\mathfrak {h}}}
2039:{\displaystyle {\mathfrak {sl}}_{2n}(\mathbb {C} )}
4301:
4223:, vol. 126 (2nd ed.), Berlin, New York:
4049:
4005:
3962:
3937:
3915:
3876:
3842:
3802:
3776:
3745:
3717:
3696:
3672:
3633:
3612:
3584:
3564:
3525:
3465:
3442:
3405:
3294:
3237:
3216:
3176:
3118:
3087:
3058:
2986:. The above decomposition can then be written as:
2978:
2954:
2930:
2885:
2865:
2769:
2651:
2580:
2556:
2513:
2477:
2414:
2387:
2359:
2273:
2253:
2233:
2213:
2160:
2141:but has a maximal abelian subalgebra of dimension
2133:
2104:
2084:
2061:
2038:
1987:
1967:
1947:
1928:{\displaystyle {\mathfrak {sl}}_{2}(\mathbb {R} )}
1927:
1879:
1790:{\displaystyle {\mathfrak {sl}}_{2}(\mathbb {C} )}
1789:
1744:
1580:
1443:{\displaystyle {\mathfrak {sl}}_{n}(\mathbb {C} )}
1442:
1397:
1366:
1339:
1293:
1277:as a linear Lie algebra, so that a subalgebra of
1269:
1241:
1217:
1193:
1161:
1129:
1098:
1056:
1024:
996:
927:
866:
836:
816:
799:representation theory of a semi-simple Lie algebra
785:
755:
725:
679:
652:
4050:{\displaystyle ({\mathfrak {g}},{\mathfrak {h}})}
3886:there exists a unique irreducible representation
1887:with Lie bracket given by the matrix commutator.
980:For a finite-dimensional semisimple Lie algebra
453:Representation theory of semisimple Lie algebras
3443:{\displaystyle \lambda \in {\mathfrak {h}}^{*}}
2528:These two properties say that the operators in
4347:Anthony William Knapp; David A. Vogan (1995).
2287:Cartan subalgebras of semisimple Lie algebras
599:
8:
3829:
3817:
3559:
3553:
3400:
3322:
2860:
2857:
2851:
2826:
2820:
2794:
2764:
2689:
3119:{\displaystyle {\mathfrak {g}}_{\lambda }}
1071:of the algebra, and in particular are all
606:
592:
491:Particle physics and representation theory
136:
33:
18:
4038:
4037:
4028:
4027:
4022:
3997:
3996:
3994:
3954:
3953:
3951:
3930:
3898:
3892:
3868:
3856:
3836:
3835:
3815:
3789:
3768:
3762:
3761:
3758:
3732:
3710:
3688:
3687:
3685:
3664:
3663:
3661:
3626:
3604:
3603:
3601:
3577:
3544:
3538:
3517:
3505:
3499:
3498:
3490:
3478:
3458:
3434:
3428:
3427:
3418:
3394:
3393:
3382:
3313:
3307:
3274:
3273:
3264:
3263:
3255:
3230:
3208:
3207:
3205:
3159:
3158:
3143:
3142:
3134:
3110:
3104:
3103:
3100:
3074:
3045:
3039:
3038:
3025:
3007:
3006:
2997:
2996:
2994:
2970:
2969:
2967:
2946:
2945:
2943:
2922:
2921:
2912:
2906:
2905:
2902:
2878:
2842:
2836:
2835:
2829:
2811:
2805:
2804:
2786:
2758:
2757:
2746:
2708:
2699:
2698:
2680:
2674:
2673:
2670:
2643:
2637:
2636:
2627:
2621:
2620:
2612:
2599:
2598:
2596:
2572:
2571:
2569:
2545:
2544:
2533:
2502:
2501:
2490:
2466:
2465:
2453:
2452:
2443:
2442:
2434:
2406:
2405:
2403:
2379:
2378:
2376:
2371:of characteristic 0, a Cartan subalgebra
2351:
2350:
2348:
2266:
2246:
2226:
2175:
2173:
2152:
2146:
2117:
2097:
2074:
2051:
2029:
2028:
2016:
2007:
2006:
2003:
1995:has two non-conjugate Cartan subalgebras.
1980:
1960:
1940:
1918:
1917:
1908:
1899:
1898:
1895:
1868:
1867:
1819:
1805:
1804:
1802:
1780:
1779:
1770:
1761:
1760:
1757:
1728:
1648:
1636:
1624:
1605:
1593:
1561:
1551:
1540:
1531:
1527:
1526:
1517:
1501:
1482:
1458:
1457:
1455:
1433:
1432:
1423:
1414:
1413:
1410:
1384:
1379:For the special Lie algebra of traceless
1353:
1331:
1322:
1321:
1318:
1285:
1284:
1282:
1261:
1260:
1258:
1233:
1232:
1230:
1209:
1208:
1206:
1185:
1184:
1182:
1153:
1152:
1150:
1121:
1120:
1118:
1090:
1089:
1087:
1048:
1047:
1045:
1016:
1015:
1013:
988:
987:
985:
919:
918:
909:
908:
888:
860:
859:
857:
829:
808:
807:
805:
797:in his doctoral thesis. It controls the
777:
776:
768:
747:
746:
738:
717:
716:
696:
671:
670:
668:
644:
643:
641:
4133:
3680:. For a finite dimensional irreducible
2817:
2293:Semisimple Lie algebra § Structure
2168:consisting of all matrices of the form
458:Representations of classical Lie groups
190:
139:
21:
4195:Lie algebras and their Representations
3753:with respect to a partial ordering on
3565:{\displaystyle V_{\lambda }\neq \{0\}}
16:Nilpotent subalgebra of a Lie algebra
7:
4178:
4139:
4137:
3877:{\displaystyle \alpha \in \Phi ^{+}}
1340:{\displaystyle {\mathfrak {gl}}_{n}}
786:{\displaystyle Y\in {\mathfrak {h}}}
756:{\displaystyle X\in {\mathfrak {h}}}
311:Lie groupâLie algebra correspondence
4039:
4029:
3998:
3955:
3777:{\displaystyle {\mathfrak {h}}^{*}}
3763:
3689:
3665:
3605:
3500:
3429:
3395:
3278:
3275:
3265:
3209:
3160:
3144:
3105:
3040:
3008:
2998:
2971:
2947:
2923:
2907:
2837:
2806:
2759:
2700:
2675:
2638:
2622:
2600:
2573:
2546:
2503:
2467:
2457:
2454:
2444:
2407:
2380:
2352:
2011:
2008:
1903:
1900:
1806:
1765:
1762:
1459:
1418:
1415:
1326:
1323:
1286:
1262:
1234:
1210:
1186:
1154:
1122:
1091:
1049:
1017:
989:
946:In general, a subalgebra is called
920:
910:
809:
778:
748:
726:{\displaystyle \in {\mathfrak {h}}}
718:
672:
645:
4123:) consisting of diagonal matrices.
3932:
3865:
3797:
3740:
3171:
3168:
3082:
3032:
2880:
2788:
14:
4200:Infinite-dimensional Lie algebras
3803:{\displaystyle \lambda \in \Phi }
3746:{\displaystyle \lambda \in \Phi }
3088:{\displaystyle \lambda \in \Phi }
3190:Semisimple Lie algebra#Structure
2300:
2112:has a Cartan subalgebra of rank
4006:{\displaystyle {\mathfrak {h}}}
3963:{\displaystyle {\mathfrak {g}}}
3916:{\displaystyle L^{+}(\lambda )}
3697:{\displaystyle {\mathfrak {g}}}
3673:{\displaystyle {\mathfrak {g}}}
3613:{\displaystyle {\mathfrak {g}}}
3224:over a field of characteristic
3217:{\displaystyle {\mathfrak {g}}}
2979:{\displaystyle {\mathfrak {h}}}
2955:{\displaystyle {\mathfrak {h}}}
2581:{\displaystyle {\mathfrak {g}}}
2429:For the adjoint representation
2415:{\displaystyle {\mathfrak {h}}}
2388:{\displaystyle {\mathfrak {h}}}
2360:{\displaystyle {\mathfrak {g}}}
1450:, it has the Cartan subalgebra
1294:{\displaystyle {\mathfrak {g}}}
1270:{\displaystyle {\mathfrak {g}}}
1242:{\displaystyle {\mathfrak {g}}}
1218:{\displaystyle {\mathfrak {g}}}
1194:{\displaystyle {\mathfrak {g}}}
1162:{\displaystyle {\mathfrak {g}}}
1130:{\displaystyle {\mathfrak {g}}}
1099:{\displaystyle {\mathfrak {h}}}
1057:{\displaystyle {\mathfrak {g}}}
1025:{\displaystyle {\mathfrak {g}}}
997:{\displaystyle {\mathfrak {g}}}
824:over a field of characteristic
817:{\displaystyle {\mathfrak {g}}}
680:{\displaystyle {\mathfrak {g}}}
653:{\displaystyle {\mathfrak {h}}}
4044:
4024:
3910:
3904:
3376:
3370:
3361:
3355:
3352:
3349:
3343:
3337:
3289:
3283:
3270:
2830:
2737:
2731:
2719:
2713:
2551:
2541:
2508:
2498:
2472:
2462:
2449:
2395:has the following properties:
2033:
2025:
1922:
1914:
1784:
1776:
1630:
1598:
1507:
1475:
1437:
1429:
959:generalized KacâMoody algebras
915:
902:
896:
710:
698:
506:Galilean group representations
501:Poincaré group representations
1:
4221:Graduate Texts in Mathematics
3727:there exists a unique weight
3652:Theorem of the highest weight
2312: with: The action of the
496:Lorentz group representations
463:Theorem of the highest weight
4300:Humphreys, James E. (1972),
4110:Examples of Cartan Subgroups
3987:splitting Cartan subalgebras
867:{\displaystyle \mathbb {C} }
4336:Encyclopedia of Mathematics
4144:Hotta, R. (Ryoshi) (2008).
3981:Splitting Cartan subalgebra
3975:Splitting Cartan subalgebra
3925:This means the root system
2938:; i.e., the centralizer of
793:). They were introduced by
4392:
4071:
3978:
3649:
3249:Lie algebra representation
3126:has dimension one and so:
3069:As it turns out, for each
2369:algebraically closed field
2318:Harish-Chandra isomorphism
2316:on the algebra, as in the
2290:
1040:). A Cartan subalgebra of
448:Lie algebra representation
4091:âoften referred to as a â
3192:for further information.
1398:{\displaystyle n\times n}
1367:{\displaystyle n\times n}
3850:for every positive root
3585:{\displaystyle \lambda }
3466:{\displaystyle \lambda }
1249:is semisimple, then the
965:Existence and uniqueness
847:In a finite-dimensional
443:Lie group representation
4249:Hall, Brian C. (2015),
4217:Linear algebraic groups
4079:A Cartan subgroup of a
3452:weight space for weight
2340:For finite-dimensional
1313:A Cartan subalgebra of
626:, often abbreviated as
468:BorelâWeilâBott theorem
4051:
4007:
3964:
3939:
3917:
3878:
3844:
3804:
3778:
3747:
3719:
3698:
3674:
3635:
3614:
3586:
3566:
3533:In addition, whenever
3527:
3467:
3444:
3407:
3296:
3239:
3218:
3178:
3120:
3089:
3060:
2980:
2956:
2932:
2887:
2867:
2771:
2653:
2582:
2558:
2515:
2479:
2416:
2389:
2361:
2342:semisimple Lie algebra
2275:
2255:
2235:
2215:
2162:
2135:
2106:
2086:
2063:
2040:
1989:
1969:
1949:
1929:
1881:
1791:
1746:
1582:
1556:
1444:
1399:
1368:
1341:
1295:
1271:
1251:adjoint representation
1243:
1219:
1195:
1163:
1131:
1110:of the compact group.
1100:
1058:
1026:
998:
929:
868:
849:semisimple Lie algebra
838:
818:
787:
757:
727:
681:
654:
366:Semisimple Lie algebra
321:Adjoint representation
4052:
4008:
3965:
3940:
3938:{\displaystyle \Phi }
3918:
3879:
3845:
3805:
3779:
3748:
3720:
3699:
3675:
3636:
3615:
3587:
3567:
3528:
3468:
3445:
3408:
3297:
3240:
3219:
3179:
3121:
3090:
3061:
2981:
2957:
2933:
2888:
2886:{\displaystyle \Phi }
2868:
2772:
2654:
2583:
2559:
2516:
2480:
2417:
2390:
2362:
2276:
2256:
2236:
2216:
2163:
2161:{\displaystyle n^{2}}
2136:
2107:
2087:
2064:
2041:
1990:
1970:
1950:
1930:
1882:
1792:
1747:
1583:
1536:
1445:
1400:
1369:
1347:, the Lie algebra of
1342:
1296:
1272:
1244:
1220:
1196:
1164:
1132:
1101:
1059:
1027:
999:
930:
869:
839:
819:
788:
758:
728:
682:
655:
435:Representation theory
4308:, Berlin, New York:
4021:
3993:
3950:
3929:
3891:
3855:
3814:
3788:
3784:. Moreover, given a
3757:
3731:
3709:
3684:
3660:
3625:
3600:
3576:
3537:
3477:
3457:
3417:
3306:
3254:
3229:
3204:
3200:Given a Lie algebra
3133:
3099:
3073:
2993:
2966:
2942:
2901:
2877:
2785:
2669:
2595:
2568:
2532:
2489:
2433:
2402:
2375:
2347:
2265:
2245:
2225:
2172:
2145:
2134:{\displaystyle 2n-1}
2116:
2096:
2073:
2050:
2002:
1979:
1959:
1939:
1894:
1801:
1756:
1592:
1454:
1409:
1383:
1352:
1317:
1281:
1257:
1229:
1205:
1181:
1149:
1117:
1086:
1044:
1034:adjoint endomorphism
1012:
984:
887:
882:adjoint endomorphism
856:
828:
804:
767:
737:
695:
667:
640:
4331:"Cartan subalgebra"
1008:is a subalgebra of
580:Table of Lie groups
421:Compact Lie algebra
4276:Dover Publications
4115:The subgroup in GL
4047:
4013:then it is called
4003:
3960:
3935:
3913:
3874:
3840:
3800:
3774:
3743:
3715:
3694:
3670:
3631:
3610:
3582:
3562:
3523:
3512:
3463:
3440:
3403:
3292:
3235:
3214:
3174:
3116:
3085:
3056:
3036:
2976:
2952:
2928:
2883:
2863:
2767:
2649:
2634:
2578:
2554:
2511:
2475:
2412:
2385:
2357:
2320:. You can help by
2271:
2251:
2231:
2211:
2205:
2158:
2131:
2102:
2092:matrices of trace
2085:{\displaystyle 2n}
2082:
2062:{\displaystyle 2n}
2059:
2036:
1985:
1975:matrices of trace
1965:
1945:
1925:
1877:
1852:
1787:
1742:
1736:
1578:
1440:
1395:
1364:
1337:
1291:
1267:
1239:
1215:
1191:
1159:
1139:linear Lie algebra
1127:
1096:
1054:
1022:
994:
955:KacâMoody algebras
925:
864:
834:
814:
783:
753:
723:
677:
650:
352:Affine Lie algebra
342:Simple Lie algebra
83:Special orthogonal
4358:978-0-691-03756-1
4319:978-0-387-90053-7
4285:978-0-486-63832-4
4234:978-0-387-97370-8
4157:978-0-8176-4363-8
4059:split Lie algebra
3718:{\displaystyle V}
3634:{\displaystyle V}
3486:
3385:
3238:{\displaystyle 0}
3021:
2749:
2711:
2608:
2338:
2337:
2274:{\displaystyle n}
2254:{\displaystyle n}
2234:{\displaystyle A}
2105:{\displaystyle 0}
1988:{\displaystyle 0}
1968:{\displaystyle 2}
1948:{\displaystyle 2}
1534:
1036:induced by it is
837:{\displaystyle 0}
624:Cartan subalgebra
616:
615:
416:Split Lie algebra
379:Cartan subalgebra
241:
240:
132:Simple Lie groups
4383:
4362:
4343:
4322:
4307:
4296:
4268:Jacobson, Nathan
4263:
4245:
4182:
4176:
4170:
4169:
4141:
4089:Abelian subgroup
4056:
4054:
4053:
4048:
4043:
4042:
4033:
4032:
4012:
4010:
4009:
4004:
4002:
4001:
3971:
3969:
3967:
3966:
3961:
3959:
3958:
3944:
3942:
3941:
3936:
3924:
3922:
3920:
3919:
3914:
3903:
3902:
3885:
3883:
3881:
3880:
3875:
3873:
3872:
3849:
3847:
3846:
3841:
3839:
3809:
3807:
3806:
3801:
3783:
3781:
3780:
3775:
3773:
3772:
3767:
3766:
3752:
3750:
3749:
3744:
3726:
3724:
3722:
3721:
3716:
3704:-representation
3703:
3701:
3700:
3695:
3693:
3692:
3679:
3677:
3676:
3671:
3669:
3668:
3642:
3640:
3638:
3637:
3632:
3620:-representation
3619:
3617:
3616:
3611:
3609:
3608:
3591:
3589:
3588:
3583:
3571:
3569:
3568:
3563:
3549:
3548:
3532:
3530:
3529:
3524:
3522:
3521:
3511:
3510:
3509:
3504:
3503:
3472:
3470:
3469:
3464:
3449:
3447:
3446:
3441:
3439:
3438:
3433:
3432:
3412:
3410:
3409:
3404:
3399:
3398:
3386:
3383:
3318:
3317:
3301:
3299:
3298:
3293:
3282:
3281:
3269:
3268:
3246:
3244:
3242:
3241:
3236:
3223:
3221:
3220:
3215:
3213:
3212:
3183:
3181:
3180:
3175:
3164:
3163:
3148:
3147:
3125:
3123:
3122:
3117:
3115:
3114:
3109:
3108:
3094:
3092:
3091:
3086:
3065:
3063:
3062:
3057:
3055:
3051:
3050:
3049:
3044:
3043:
3035:
3012:
3011:
3002:
3001:
2985:
2983:
2982:
2977:
2975:
2974:
2961:
2959:
2958:
2953:
2951:
2950:
2937:
2935:
2934:
2929:
2927:
2926:
2917:
2916:
2911:
2910:
2892:
2890:
2889:
2884:
2872:
2870:
2869:
2864:
2847:
2846:
2841:
2840:
2833:
2816:
2815:
2810:
2809:
2776:
2774:
2773:
2768:
2763:
2762:
2750:
2747:
2712:
2709:
2704:
2703:
2685:
2684:
2679:
2678:
2658:
2656:
2655:
2650:
2648:
2647:
2642:
2641:
2633:
2632:
2631:
2626:
2625:
2604:
2603:
2587:
2585:
2584:
2579:
2577:
2576:
2563:
2561:
2560:
2555:
2550:
2549:
2520:
2518:
2517:
2512:
2507:
2506:
2484:
2482:
2481:
2476:
2471:
2470:
2461:
2460:
2448:
2447:
2421:
2419:
2418:
2413:
2411:
2410:
2394:
2392:
2391:
2386:
2384:
2383:
2366:
2364:
2363:
2358:
2356:
2355:
2333:
2330:
2304:
2297:
2280:
2278:
2277:
2272:
2260:
2258:
2257:
2252:
2240:
2238:
2237:
2232:
2220:
2218:
2217:
2212:
2210:
2209:
2167:
2165:
2164:
2159:
2157:
2156:
2140:
2138:
2137:
2132:
2111:
2109:
2108:
2103:
2091:
2089:
2088:
2083:
2068:
2066:
2065:
2060:
2045:
2043:
2042:
2037:
2032:
2024:
2023:
2015:
2014:
1994:
1992:
1991:
1986:
1974:
1972:
1971:
1966:
1954:
1952:
1951:
1946:
1934:
1932:
1931:
1926:
1921:
1913:
1912:
1907:
1906:
1890:The Lie algebra
1886:
1884:
1883:
1878:
1876:
1872:
1871:
1857:
1856:
1810:
1809:
1796:
1794:
1793:
1788:
1783:
1775:
1774:
1769:
1768:
1752:For example, in
1751:
1749:
1748:
1743:
1741:
1740:
1733:
1732:
1695:
1682:
1653:
1652:
1629:
1628:
1610:
1609:
1587:
1585:
1584:
1579:
1577:
1573:
1566:
1565:
1555:
1550:
1535:
1532:
1530:
1522:
1521:
1506:
1505:
1487:
1486:
1463:
1462:
1449:
1447:
1446:
1441:
1436:
1428:
1427:
1422:
1421:
1404:
1402:
1401:
1396:
1373:
1371:
1370:
1365:
1346:
1344:
1343:
1338:
1336:
1335:
1330:
1329:
1300:
1298:
1297:
1292:
1290:
1289:
1276:
1274:
1273:
1268:
1266:
1265:
1248:
1246:
1245:
1240:
1238:
1237:
1224:
1222:
1221:
1216:
1214:
1213:
1200:
1198:
1197:
1192:
1190:
1189:
1175:toral subalgebra
1168:
1166:
1165:
1160:
1158:
1157:
1136:
1134:
1133:
1128:
1126:
1125:
1105:
1103:
1102:
1097:
1095:
1094:
1079:of the algebra.
1063:
1061:
1060:
1055:
1053:
1052:
1031:
1029:
1028:
1023:
1021:
1020:
1006:toral subalgebra
1003:
1001:
1000:
995:
993:
992:
934:
932:
931:
926:
924:
923:
914:
913:
875:
873:
871:
870:
865:
863:
843:
841:
840:
835:
823:
821:
820:
815:
813:
812:
792:
790:
789:
784:
782:
781:
762:
760:
759:
754:
752:
751:
732:
730:
729:
724:
722:
721:
689:self-normalising
686:
684:
683:
678:
676:
675:
659:
657:
656:
651:
649:
648:
608:
601:
594:
551:Claude Chevalley
408:Complexification
251:Other Lie groups
137:
45:Classical groups
37:
19:
4391:
4390:
4386:
4385:
4384:
4382:
4381:
4380:
4366:
4365:
4359:
4346:
4325:
4320:
4310:Springer-Verlag
4299:
4286:
4266:
4261:
4248:
4235:
4225:Springer-Verlag
4211:
4208:
4191:
4186:
4185:
4177:
4173:
4158:
4143:
4142:
4135:
4130:
4118:
4112:
4077:
4074:Cartan subgroup
4070:
4068:Cartan subgroup
4019:
4018:
3991:
3990:
3983:
3977:
3948:
3947:
3946:
3927:
3926:
3894:
3889:
3888:
3887:
3864:
3853:
3852:
3851:
3812:
3811:
3786:
3785:
3760:
3755:
3754:
3729:
3728:
3707:
3706:
3705:
3682:
3681:
3658:
3657:
3654:
3648:
3623:
3622:
3621:
3598:
3597:
3574:
3573:
3540:
3535:
3534:
3513:
3497:
3475:
3474:
3455:
3454:
3426:
3415:
3414:
3384: for
3309:
3304:
3303:
3252:
3251:
3227:
3226:
3225:
3202:
3201:
3198:
3131:
3130:
3102:
3097:
3096:
3071:
3070:
3037:
3020:
3016:
2991:
2990:
2964:
2963:
2962:coincides with
2940:
2939:
2904:
2899:
2898:
2897:and, moreover,
2875:
2874:
2834:
2803:
2783:
2782:
2748: for
2672:
2667:
2666:
2635:
2619:
2593:
2592:
2566:
2565:
2530:
2529:
2487:
2486:
2431:
2430:
2400:
2399:
2373:
2372:
2345:
2344:
2334:
2328:
2325:
2310:needs expansion
2295:
2289:
2263:
2262:
2243:
2242:
2223:
2222:
2204:
2203:
2198:
2192:
2191:
2186:
2176:
2170:
2169:
2148:
2143:
2142:
2114:
2113:
2094:
2093:
2071:
2070:
2048:
2047:
2005:
2000:
1999:
1977:
1976:
1957:
1956:
1937:
1936:
1897:
1892:
1891:
1851:
1850:
1842:
1836:
1835:
1830:
1820:
1818:
1814:
1799:
1798:
1759:
1754:
1753:
1735:
1734:
1724:
1722:
1717:
1712:
1706:
1705:
1700:
1694:
1688:
1687:
1681:
1676:
1670:
1669:
1664:
1659:
1654:
1644:
1637:
1620:
1601:
1590:
1589:
1557:
1533: and
1513:
1497:
1478:
1471:
1467:
1452:
1451:
1412:
1407:
1406:
1381:
1380:
1350:
1349:
1320:
1315:
1314:
1307:
1279:
1278:
1255:
1254:
1227:
1226:
1203:
1202:
1179:
1178:
1147:
1146:
1115:
1114:
1084:
1083:
1042:
1041:
1010:
1009:
982:
981:
975:regular element
967:
885:
884:
854:
853:
852:
826:
825:
802:
801:
765:
764:
735:
734:
693:
692:
665:
664:
638:
637:
612:
567:
566:
565:
536:Wilhelm Killing
520:
512:
511:
510:
485:
474:
473:
472:
437:
427:
426:
425:
412:
396:
374:Dynkin diagrams
368:
358:
357:
356:
338:
316:Exponential map
305:
295:
294:
293:
274:Conformal group
253:
243:
242:
234:
226:
218:
210:
202:
183:
173:
163:
153:
134:
124:
123:
122:
103:Special unitary
47:
17:
12:
11:
5:
4389:
4387:
4379:
4378:
4368:
4367:
4364:
4363:
4357:
4344:
4323:
4318:
4297:
4284:
4264:
4260:978-3319134666
4259:
4246:
4233:
4207:
4204:
4203:
4202:
4197:
4190:
4187:
4184:
4183:
4171:
4156:
4132:
4131:
4129:
4126:
4125:
4124:
4116:
4111:
4108:
4069:
4066:
4046:
4041:
4036:
4031:
4026:
4000:
3979:Main article:
3976:
3973:
3957:
3934:
3912:
3909:
3906:
3901:
3897:
3871:
3867:
3863:
3860:
3838:
3834:
3831:
3828:
3825:
3822:
3819:
3799:
3796:
3793:
3771:
3765:
3742:
3739:
3736:
3714:
3691:
3667:
3650:Main article:
3647:
3644:
3630:
3607:
3581:
3561:
3558:
3555:
3552:
3547:
3543:
3520:
3516:
3508:
3502:
3496:
3493:
3489:
3485:
3482:
3462:
3437:
3431:
3425:
3422:
3402:
3397:
3392:
3389:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3316:
3312:
3291:
3288:
3285:
3280:
3277:
3272:
3267:
3262:
3259:
3234:
3211:
3197:
3194:
3186:
3185:
3173:
3170:
3167:
3162:
3157:
3154:
3151:
3146:
3141:
3138:
3113:
3107:
3084:
3081:
3078:
3067:
3066:
3054:
3048:
3042:
3034:
3031:
3028:
3024:
3019:
3015:
3010:
3005:
3000:
2973:
2949:
2925:
2920:
2915:
2909:
2882:
2862:
2859:
2856:
2853:
2850:
2845:
2839:
2832:
2828:
2825:
2822:
2819:
2814:
2808:
2802:
2799:
2796:
2793:
2790:
2779:
2778:
2766:
2761:
2756:
2753:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2707:
2702:
2697:
2694:
2691:
2688:
2683:
2677:
2660:
2659:
2646:
2640:
2630:
2624:
2618:
2615:
2611:
2607:
2602:
2575:
2553:
2548:
2543:
2540:
2537:
2523:
2522:
2510:
2505:
2500:
2497:
2494:
2474:
2469:
2464:
2459:
2456:
2451:
2446:
2441:
2438:
2427:
2409:
2382:
2354:
2336:
2335:
2307:
2305:
2288:
2285:
2284:
2283:
2270:
2250:
2230:
2208:
2202:
2199:
2197:
2194:
2193:
2190:
2187:
2185:
2182:
2181:
2179:
2155:
2151:
2130:
2127:
2124:
2121:
2101:
2081:
2078:
2058:
2055:
2035:
2031:
2027:
2022:
2019:
2013:
2010:
1996:
1984:
1964:
1944:
1924:
1920:
1916:
1911:
1905:
1902:
1888:
1875:
1870:
1866:
1863:
1860:
1855:
1849:
1846:
1843:
1841:
1838:
1837:
1834:
1831:
1829:
1826:
1825:
1823:
1817:
1813:
1808:
1786:
1782:
1778:
1773:
1767:
1764:
1739:
1731:
1727:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1707:
1704:
1701:
1699:
1696:
1693:
1690:
1689:
1686:
1683:
1680:
1677:
1675:
1672:
1671:
1668:
1665:
1663:
1660:
1658:
1655:
1651:
1647:
1643:
1642:
1640:
1635:
1632:
1627:
1623:
1619:
1616:
1613:
1608:
1604:
1600:
1597:
1576:
1572:
1569:
1564:
1560:
1554:
1549:
1546:
1543:
1539:
1529:
1525:
1520:
1516:
1512:
1509:
1504:
1500:
1496:
1493:
1490:
1485:
1481:
1477:
1474:
1470:
1466:
1461:
1439:
1435:
1431:
1426:
1420:
1417:
1394:
1391:
1388:
1377:
1363:
1360:
1357:
1334:
1328:
1325:
1311:
1306:
1303:
1288:
1264:
1236:
1212:
1188:
1156:
1124:
1093:
1051:
1038:diagonalizable
1019:
991:
966:
963:
941:diagonalizable
922:
917:
912:
907:
904:
901:
898:
895:
892:
880:such that the
862:
833:
811:
780:
775:
772:
750:
745:
742:
720:
715:
712:
709:
706:
703:
700:
674:
647:
614:
613:
611:
610:
603:
596:
588:
585:
584:
583:
582:
577:
569:
568:
564:
563:
558:
556:Harish-Chandra
553:
548:
543:
538:
533:
531:Henri Poincaré
528:
522:
521:
518:
517:
514:
513:
509:
508:
503:
498:
493:
487:
486:
481:Lie groups in
480:
479:
476:
475:
471:
470:
465:
460:
455:
450:
445:
439:
438:
433:
432:
429:
428:
424:
423:
418:
413:
411:
410:
405:
399:
397:
395:
394:
389:
383:
381:
376:
370:
369:
364:
363:
360:
359:
355:
354:
349:
344:
339:
337:
336:
331:
325:
323:
318:
313:
307:
306:
301:
300:
297:
296:
292:
291:
286:
281:
279:Diffeomorphism
276:
271:
266:
261:
255:
254:
249:
248:
245:
244:
239:
238:
237:
236:
232:
228:
224:
220:
216:
212:
208:
204:
200:
193:
192:
188:
187:
186:
185:
179:
175:
169:
165:
159:
155:
149:
142:
141:
135:
130:
129:
126:
125:
121:
120:
110:
100:
90:
80:
70:
63:Special linear
60:
53:General linear
49:
48:
43:
42:
39:
38:
30:
29:
15:
13:
10:
9:
6:
4:
3:
2:
4388:
4377:
4374:
4373:
4371:
4360:
4354:
4350:
4345:
4342:
4338:
4337:
4332:
4328:
4324:
4321:
4315:
4311:
4306:
4305:
4298:
4295:
4291:
4287:
4281:
4277:
4273:
4269:
4265:
4262:
4256:
4252:
4247:
4244:
4240:
4236:
4230:
4226:
4222:
4218:
4214:
4213:Borel, Armand
4210:
4209:
4205:
4201:
4198:
4196:
4193:
4192:
4188:
4180:
4175:
4172:
4167:
4163:
4159:
4153:
4149:
4148:
4140:
4138:
4134:
4127:
4122:
4114:
4113:
4109:
4107:
4105:
4101:
4096:
4094:
4093:maximal torus
4090:
4085:
4082:
4075:
4067:
4065:
4062:
4060:
4034:
4017:and the pair
4016:
3988:
3982:
3974:
3972:
3907:
3899:
3895:
3869:
3861:
3858:
3832:
3826:
3823:
3820:
3794:
3791:
3769:
3737:
3734:
3712:
3653:
3645:
3643:
3628:
3595:
3579:
3556:
3550:
3545:
3541:
3518:
3514:
3506:
3494:
3491:
3487:
3483:
3480:
3460:
3453:
3450:, called the
3435:
3423:
3420:
3390:
3387:
3379:
3373:
3367:
3364:
3358:
3346:
3340:
3334:
3331:
3328:
3325:
3319:
3314:
3310:
3286:
3260:
3257:
3250:
3232:
3195:
3193:
3191:
3165:
3155:
3152:
3149:
3139:
3136:
3129:
3128:
3127:
3111:
3079:
3076:
3052:
3046:
3029:
3026:
3022:
3017:
3013:
3003:
2989:
2988:
2987:
2918:
2913:
2896:
2854:
2848:
2843:
2823:
2812:
2800:
2797:
2791:
2754:
2751:
2743:
2740:
2734:
2728:
2725:
2722:
2716:
2705:
2695:
2692:
2686:
2681:
2665:
2664:
2663:
2644:
2628:
2616:
2613:
2609:
2605:
2591:
2590:
2589:
2538:
2535:
2526:
2495:
2492:
2439:
2436:
2428:
2425:
2398:
2397:
2396:
2370:
2343:
2332:
2329:February 2014
2323:
2319:
2315:
2311:
2308:This section
2306:
2303:
2299:
2298:
2294:
2286:
2268:
2248:
2228:
2206:
2200:
2195:
2188:
2183:
2177:
2153:
2149:
2128:
2125:
2122:
2119:
2099:
2079:
2076:
2056:
2053:
2020:
2017:
1997:
1982:
1962:
1942:
1909:
1889:
1873:
1864:
1861:
1858:
1853:
1847:
1844:
1839:
1832:
1827:
1821:
1815:
1811:
1771:
1737:
1729:
1725:
1719:
1714:
1709:
1702:
1697:
1691:
1684:
1678:
1673:
1666:
1661:
1656:
1649:
1645:
1638:
1633:
1625:
1621:
1617:
1614:
1611:
1606:
1602:
1595:
1574:
1570:
1567:
1562:
1558:
1552:
1547:
1544:
1541:
1537:
1523:
1518:
1514:
1510:
1502:
1498:
1494:
1491:
1488:
1483:
1479:
1472:
1468:
1464:
1424:
1392:
1389:
1386:
1378:
1375:
1361:
1358:
1355:
1332:
1312:
1309:
1308:
1304:
1302:
1252:
1176:
1173:of a maximal
1172:
1144:
1140:
1111:
1109:
1108:maximal torus
1080:
1078:
1074:
1070:
1069:automorphisms
1065:
1039:
1035:
1007:
978:
976:
972:
964:
962:
960:
956:
952:
949:
944:
942:
938:
905:
899:
893:
890:
883:
879:
850:
845:
831:
800:
796:
773:
770:
743:
740:
713:
707:
704:
701:
690:
663:
636:
633:
629:
625:
621:
609:
604:
602:
597:
595:
590:
589:
587:
586:
581:
578:
576:
573:
572:
571:
570:
562:
559:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
532:
529:
527:
524:
523:
516:
515:
507:
504:
502:
499:
497:
494:
492:
489:
488:
484:
478:
477:
469:
466:
464:
461:
459:
456:
454:
451:
449:
446:
444:
441:
440:
436:
431:
430:
422:
419:
417:
414:
409:
406:
404:
401:
400:
398:
393:
390:
388:
385:
384:
382:
380:
377:
375:
372:
371:
367:
362:
361:
353:
350:
348:
345:
343:
340:
335:
332:
330:
327:
326:
324:
322:
319:
317:
314:
312:
309:
308:
304:
299:
298:
290:
287:
285:
282:
280:
277:
275:
272:
270:
267:
265:
262:
260:
257:
256:
252:
247:
246:
235:
229:
227:
221:
219:
213:
211:
205:
203:
197:
196:
195:
194:
189:
184:
182:
176:
174:
172:
166:
164:
162:
156:
154:
152:
146:
145:
144:
143:
138:
133:
128:
127:
118:
114:
111:
108:
104:
101:
98:
94:
91:
88:
84:
81:
78:
74:
71:
68:
64:
61:
58:
54:
51:
50:
46:
41:
40:
36:
32:
31:
28:
24:
20:
4376:Lie algebras
4348:
4334:
4303:
4274:, New York:
4272:Lie algebras
4271:
4250:
4216:
4174:
4146:
4120:
4104:Weyl chamber
4097:
4086:
4078:
4063:
4057:is called a
4014:
3984:
3655:
3593:
3451:
3199:
3187:
3068:
2780:
2661:
2527:
2524:
2485:, the image
2339:
2326:
2322:adding to it
2309:
1142:
1112:
1081:
1066:
979:
968:
953:
945:
877:
846:
627:
623:
617:
561:Armand Borel
546:Hermann Weyl
378:
347:Loop algebra
329:Killing form
303:Lie algebras
180:
170:
160:
150:
116:
106:
96:
86:
76:
66:
56:
27:Lie algebras
4327:Popov, V.L.
4100:David Vogan
4015:splittable,
2895:root system
1171:centralizer
795:Ălie Cartan
662:Lie algebra
620:mathematics
541:Ălie Cartan
387:Root system
191:Exceptional
4206:References
4128:References
3810:such that
2314:Weyl group
2291:See also:
2282:matrices).
1073:isomorphic
937:semisimple
635:subalgebra
526:Sophus Lie
519:Scientists
392:Weyl group
113:Symplectic
73:Orthogonal
23:Lie groups
4341:EMS Press
4329:(2001) ,
4181:Chapter 7
4179:Hall 2015
4166:316693861
4081:Lie group
3933:Φ
3908:λ
3866:Φ
3862:∈
3859:α
3833:∈
3830:⟩
3827:λ
3821:α
3818:⟨
3798:Φ
3795:∈
3792:λ
3770:∗
3741:Φ
3738:∈
3735:λ
3580:λ
3551:≠
3546:λ
3519:λ
3507:∗
3495:∈
3492:λ
3488:⨁
3461:λ
3436:∗
3424:∈
3421:λ
3391:∈
3368:λ
3341:σ
3329:∈
3315:λ
3271:→
3258:σ
3188:See also
3172:Φ
3169:#
3156:
3140:
3112:λ
3083:Φ
3080:∈
3077:λ
3047:λ
3033:Φ
3030:∈
3027:λ
3023:⨁
3014:⊕
2881:Φ
2849:≠
2844:λ
2818:∖
2813:∗
2801:∈
2798:λ
2789:Φ
2755:∈
2729:λ
2696:∈
2682:λ
2645:λ
2629:∗
2617:∈
2614:λ
2610:⨁
2539:
2496:
2450:→
2126:−
1865:∈
1845:−
1720:⋯
1715:⋯
1703:⋮
1698:⋱
1692:⋮
1679:⋱
1662:⋯
1615:…
1538:∑
1524:∈
1511:∣
1492:…
1405:matrices
1390:×
1359:×
1253:presents
916:→
894:
774:∈
744:∈
714:∈
632:nilpotent
403:Real form
289:Euclidean
140:Classical
4370:Category
4270:(1979),
4215:(1991),
3572:we call
2367:over an
1374:matrices
1305:Examples
733:for all
687:that is
575:Glossary
269:Poincaré
4294:0559927
4243:1102012
3596:of the
2873:. Then
2662:where
2424:abelian
1169:is the
939:(i.e.,
763:, then
630:, is a
483:physics
264:Lorentz
93:Unitary
4355:
4316:
4292:
4282:
4257:
4241:
4231:
4164:
4154:
3594:weight
3247:and a
1588:where
259:Circle
4189:Notes
3413:with
2893:is a
2221:with
1201:. If
1137:is a
971:field
948:toral
660:of a
622:, a
334:Index
4353:ISBN
4314:ISBN
4280:ISBN
4255:ISBN
4229:ISBN
4162:OCLC
4152:ISBN
2781:Let
2241:any
1077:rank
957:and
691:(if
284:Loop
25:and
3153:dim
3137:dim
2588:as
2422:is
2324:.
2261:by
2069:by
2046:of
1955:by
1935:of
1177:of
1113:If
935:is
628:CSA
618:In
115:Sp(
105:SU(
85:SO(
65:SL(
55:GL(
4372::
4351:.
4339:,
4333:,
4312:,
4290:MR
4288:,
4278:,
4239:MR
4237:,
4227:,
4219:,
4160:.
4136:^
3592:a
3095:,
2710:ad
2536:ad
2493:ad
2437:ad
891:ad
874:),
844:.
95:U(
75:O(
4361:.
4168:.
4121:R
4119:(
4117:2
4076:.
4045:)
4040:h
4035:,
4030:g
4025:(
3999:h
3970:.
3956:g
3923:.
3911:)
3905:(
3900:+
3896:L
3884:,
3870:+
3837:N
3824:,
3764:h
3725:,
3713:V
3690:g
3666:g
3641:.
3629:V
3606:g
3560:}
3557:0
3554:{
3542:V
3515:V
3501:h
3484:=
3481:V
3430:h
3401:}
3396:h
3388:h
3380:v
3377:)
3374:h
3371:(
3365:=
3362:)
3359:v
3356:(
3353:)
3350:)
3347:h
3344:(
3338:(
3335::
3332:V
3326:v
3323:{
3320:=
3311:V
3290:)
3287:V
3284:(
3279:l
3276:g
3266:g
3261::
3245:,
3233:0
3210:g
3184:.
3166:+
3161:h
3150:=
3145:g
3106:g
3053:)
3041:g
3018:(
3009:h
3004:=
2999:g
2972:h
2948:h
2924:h
2919:=
2914:0
2908:g
2861:}
2858:}
2855:0
2852:{
2838:g
2831:|
2827:}
2824:0
2821:{
2807:h
2795:{
2792:=
2777:.
2765:}
2760:h
2752:h
2744:,
2741:x
2738:)
2735:h
2732:(
2726:=
2723:x
2720:)
2717:h
2714:(
2706::
2701:g
2693:x
2690:{
2687:=
2676:g
2639:g
2623:h
2606:=
2601:g
2574:g
2552:)
2547:h
2542:(
2509:)
2504:h
2499:(
2473:)
2468:g
2463:(
2458:l
2455:g
2445:g
2440::
2426:,
2408:h
2381:h
2353:g
2331:)
2327:(
2269:n
2249:n
2229:A
2207:)
2201:0
2196:0
2189:A
2184:0
2178:(
2154:2
2150:n
2129:1
2123:n
2120:2
2100:0
2080:n
2077:2
2057:n
2054:2
2034:)
2030:C
2026:(
2021:n
2018:2
2012:l
2009:s
1983:0
1963:2
1943:2
1923:)
1919:R
1915:(
1910:2
1904:l
1901:s
1874:}
1869:C
1862:a
1859::
1854:)
1848:a
1840:0
1833:0
1828:a
1822:(
1816:{
1812:=
1807:h
1785:)
1781:C
1777:(
1772:2
1766:l
1763:s
1738:)
1730:n
1726:a
1710:0
1685:0
1674:0
1667:0
1657:0
1650:1
1646:a
1639:(
1634:=
1631:)
1626:n
1622:a
1618:,
1612:,
1607:1
1603:a
1599:(
1596:d
1575:}
1571:0
1568:=
1563:i
1559:a
1553:n
1548:1
1545:=
1542:i
1528:C
1519:i
1515:a
1508:)
1503:n
1499:a
1495:,
1489:,
1484:1
1480:a
1476:(
1473:d
1469:{
1465:=
1460:h
1438:)
1434:C
1430:(
1425:n
1419:l
1416:s
1393:n
1387:n
1362:n
1356:n
1333:n
1327:l
1324:g
1287:g
1263:g
1235:g
1211:g
1187:g
1155:g
1143:V
1123:g
1092:h
1050:g
1018:g
990:g
921:g
911:g
906::
903:)
900:x
897:(
878:x
861:C
832:0
810:g
779:h
771:Y
749:h
741:X
719:h
711:]
708:Y
705:,
702:X
699:[
673:g
646:h
607:e
600:t
593:v
233:8
231:E
225:7
223:E
217:6
215:E
209:4
207:F
201:2
199:G
181:n
178:D
171:n
168:C
161:n
158:B
151:n
148:A
119:)
117:n
109:)
107:n
99:)
97:n
89:)
87:n
79:)
77:n
69:)
67:n
59:)
57:n
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